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This problem doesn't seem very complicated but I got stuck at trying to understand what is the meaning of the last clue involving an integer and a range. Can somebody help me?

The problem is as follows:

Marina is reading a novel. The first day she read a third of the book, the second day she read the fourth parts of what it was left, the third day she read half of what it was left to read, the fourth day she read the fifth parts of what it was still left to read, the fifth day she decided to end the novel and found that it was left less than $70$ pages. If she always read an integer number of pages and never read less than $14$ pages. How many pages did the novel had?

The alternatives given in my book were as follows:

$\begin{array}{ll} 1.&\textrm{360 pages}\\ 2.&\textrm{240 pages}\\ 3.&\textrm{180 pages}\\ 4.&\textrm{300 pages}\\ 5.&\textrm{210 pages}\\ \end{array}$

I'm lost with this problem. What would be the correct way to go?. So far what I attempted to do was the following:

I thought that the number of pages that the book has to be $x$.

Since it said that the first day she read a third of the book I defined it as:

$\frac{1}{3}x$

On the second day it is said that she read the fourth parts of what it was left so that would account for:

$\frac{1}{4}\left(x-\frac{1}{3}x\right)=\frac{1}{4}\left(\frac{2x}{3}\right)=\frac{x}{6}$

The third day:

$\frac{1}{2}\left(x-\frac{x}{6}\right)=\frac{1}{2}\left(\frac{5x}{6}\right)=\frac{5x}{12}$

The fourth day:

$\frac{1}{5}\left(x-\frac{5x}{12}\right)=\frac{1}{5}\left(\frac{7x}{12}\right)=\frac{7x}{60}$

The fifth day:

She decides to end reading the novel but, what it was left was less than $70$ pages.

So this would translate as:

$x-\frac{7x}{60}<70$

This would become into:

$\frac{53x}{60}<70$

Therefore:

$53x<4200$

$x<\frac{4200}{53}$

However this fraction is not an integer.

There is also another piece of information which mentioned that she always read no less than $14$ pages.

If during the first day she read a third of the novel then this would be:

$\frac{1}{3}x>14$

So $x>42$

But, on the fourth day she read:

$\frac{7x}{60}>14$

Therefore:

$x>120$

How come x can be greatest than $42$ and at the same time $120$?. Am I understanding this correctly?.

If I were to select the greatest value and put it in the range which I found earlier:

$120<x<\frac{4200}{53}$

and round to the nearest integer:

$120<x<79$

Which doesn't make sense.

If it were $\frac{5x}{12}>14$

$x>33$ (rounded to the nearest integer)

Which would be:

$33<x<79$

But again this doesn't produce an reasonable answer within the specified range in the answers. Did I overlooked something or perhaps didn't understood something right?. Can somebody help me with this inequation problem?.

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Compute the fractions of the book read per day:

  • On day 1, $\frac13$ of the novel was read, leaving $\frac23$.
  • On day 2, $\frac23×\frac14=\frac16$ was read, leaving $\frac23×\frac34=\frac12$.
  • On day 3, $\frac12×\frac12=\frac14$ was read, the same fraction being left.
  • On day 4, $\frac14×\frac15=\frac1{20}$ was read, leaving $\frac14×\frac45=\frac15$ that was finished off on day 5.

Letting $x$ be the number of pages in the book, because at most 70 pages were left on day 5 we have $\frac15x<70$ or $x<350$. Because at least 14 pages were read per day, including day 4, we have $\frac1{20}x>14$ or $x>280$. Only option 4 satisfies both inequalities, so the novel had 300 pages.

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  • $\begingroup$ I would really like to understand what you meant. But the thing is you ommited some steps and I don't get very clearly where do those fractions come?. I don't get the idea in the second step for day 2. I understand the part of $\frac{1}{4}\times\frac{2}{3}$ but why should I multiply $\frac{2}{3} \times \frac{3}{4}$? In other words why should be multiplied what it is left instead of summing them?. $\endgroup$ – Chris Steinbeck Bell Mar 2 at 21:33
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Your method is good, but you made a mistake on the third day. Indeed, she read a third of what was left. So she read :

$$ \frac{1}{2}\left( x - \frac{1}{3}x-\frac{1}{6}x \right) = \frac{1}{2}\frac{1}{2}x = \frac{1}{4}x$$

So on the fourth day, she read $$ \frac{1}{5}\left(1 - \frac{1}{2} - \frac{1}{4}\right)x = \frac{1}{5}\frac{1}{4}x = \frac{1}{20}x$$

So on the fifth days, there's : $$ x\left(1 - \frac{1}{3} - \frac{1}{6} - \frac{1}{4} - \frac{1}{20}\right) = \frac{1}{5}{x}$$ Pages left.

To sum up :

First day : $\frac{1}{3}x$

Second day : $\frac{1}{6}x$

Third day : $\frac{1}{4}x$

Fourth day : $\frac{1}{20}x$

You want $\frac{x}{20} \ge 14$ so $x\ge 280$. Furthermore, you need $\frac{1}{5}x< 70$ so $x<350$.

The only possibility now is answer 4 : 300 pages.

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  • $\begingroup$ Thanks for the confidence boost. I really needed it. I think the source of my confusion was that I did not consider the passage mentions each day individually and not the number of pages that had been read until that day. Hence for each day I need to account what it was read on the prior day. Because of this for the third day $\frac{1}{2}\left(x-\left(\frac{x}{3}+\frac{x}{6}\right)\right)$ as $\frac{1x}{3}$ accounts for day 1 and $\frac{1x}{6}$ accounts for day two, so they must be summed up and so on until get to the last day. Did I understood this part correctly? $\endgroup$ – Chris Steinbeck Bell Mar 2 at 21:12
  • $\begingroup$ Now I get to some confusion, why do you use $\leq$ and $\geq$ with $<$ and $>$ almost interchangeably? By the time I got to $\frac{x}{5}$ it is obviously stated in the problem that is less than $70$. But the passage also mentions that everyday she read no less than $14$ pages wouldn't this meant that I can use the other found quantities as well?. Let's say $\frac{1}{4}x\geq 14$ hence $x\geq 56$ isn't it?. But also meant that on the first day $\frac{1}{3}x\geq 14$ so $x \geq 42$. I'm kind of confused, having found these inequalities don't produce contradictory results?. $\endgroup$ – Chris Steinbeck Bell Mar 2 at 21:24
  • $\begingroup$ Or just because they're inequalities it is possible to have different answers. It turns that I may need to look for the one which produces a highest number so I can reduce the boundary and find what number of pages they're asking. This problem in particular offered alternatives, but could this have been solved without any? by only concluding something from what it is given? $\endgroup$ – Chris Steinbeck Bell Mar 2 at 21:25
  • $\begingroup$ Yes you understood correctly ! I use both < and \le because I don't really care about an exact boundary, I just need a good enough one to answer the question, but there is no logic in the use of one or the other. $\endgroup$ – aleph0 Mar 3 at 20:44
  • $\begingroup$ Thanks for that but I have a question. Does it mean that given the conditions at itself it cannot be found the number of pages without checking the alternatives given?. $\endgroup$ – Chris Steinbeck Bell Mar 5 at 22:29

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